Hyperbolic Geometry for Visualizationweb.engr.oregonstate.edu/~mjb/cs553/Handouts/hyperbolic...1 mjb...
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1 mjb – February 28, 2019 Computer Graphics 1 Hyperbolic Geometry for Visualization hyperbolic.ppts Mike Bailey [email protected] mjb – February 28, 2019 Computer Graphics 2 Zooming and Panning Around a Complex 2D Display • Standard (Euclidean) geometry zooming forces much of the information off the screen • This eliminates the context from the zoomed-in display • This problem can be solved with hyperbolic methods if we are willing to give up Euclidean geometry
Transcript of Hyperbolic Geometry for Visualizationweb.engr.oregonstate.edu/~mjb/cs553/Handouts/hyperbolic...1 mjb...
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mjb – February 28, 2019Computer Graphics
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Hyperbolic Geometry for Visualization
hyperbolic.ppts
Mike [email protected]
mjb – February 28, 2019Computer Graphics
2Zooming and Panning Around a Complex 2D Display
• Standard (Euclidean) geometry zooming forces much of the information off the screen
• This eliminates the context from the zoomed-in display
• This problem can be solved with hyperbolic methods if we are willing to give up Euclidean geometry
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mjb – February 28, 2019Computer Graphics
3Usual Zooming in Euclidean Space
123,101 line strips446,585 points
mjb – February 28, 2019Computer Graphics
4Zooming in Polar Hyperbolic Space
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mjb – February 28, 2019Computer Graphics
5Polar Hyperbolic Equations
(X,Y)
R
Θ
R’ = R / (R+K)
Θ’ = Θ
X’ = R’cosΘ’Y’ = R’sinΘ’
Overall theme: something divided by something else a little bigger
1'lim0
RK
0'lim
RK
then:'RR
R K
Because
mjb – February 28, 2019Computer Graphics
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YXR 22
)(tan 1 XY
KRRR
'
KRX
RX
KRRRX
cos''
' 'sin R Y YY RR K R R K
Polar Hyperbolic Equations Don’t Actually Need to use Trig
Coordinates moved to outer edge when K = 0
Coordinates moved to center when K = ∞
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mjb – February 28, 2019Computer Graphics
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KRXX
'
Cartesian Hyperbolic Equations – Treat X and Y Independently
KRYY
'
KX
XX22
'
KY
YY22
'
{{Polar
Cartesian
Coordinates moved to outer edge when K = 0
Coordinates moved to center when K = ∞
mjb – February 28, 2019Computer Graphics
8Zooming in Cartesian Hyperbolic Space
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mjb – February 28, 2019Computer Graphics
9The Problem with T-Intersections
mjb – February 28, 2019Computer Graphics
10The Problem with T-Intersections
Your code computes the hyperbolic transformation here and here, and OpenGL draws a straight line between them. But, this point had its hyperbolic transformation computed separately, and doesn’t match up with the straight line.
This kind of situation is called a T-intersection, and crops up all the time in computer graphics, even though we don’t want it to.
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mjb – February 28, 2019Computer Graphics
11A Solution to the T-Intersection Problem
Break this line up into several (many?) sub-pieces, and perform the Hyperbolic Transformation on each intermediate point.
This makes that straight line into a curve, as it should be. But, how many line segments should we use?
0 1( ) (1 )P t t P tP t = 0., .01, .02, .03, …
mjb – February 28, 2019Computer Graphics
12A More Elegant Approach is to Recursively Subdivide
voidDrawHyperbolicLine( P0, P1 ){
Compute point
Convert point A to Hyperbolic Coordinates, calling it A’
Convert P0 and P1 to Hyperbolic Coordinates P0’, P1’
Compute point
Compare A’ and Bif( they are “close enough” ){
Draw the line P0’-P1’}else{
DrawHyperbolicLine( P0, A );DrawHyperbolicLine( A, P1 );
}}
0 1
2.P PA
0 1' ''2.
P PB
Subdividing to render a curve or surface correctly is a recurring theme in computer graphics.
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mjb – February 28, 2019Computer Graphics
13Hyperbolic Corvallis (Streets, Buildings, Parks)
Kelley Engineering
Center
mjb – February 28, 2019Computer Graphics
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http://www.sott.net/articles/show/215021-Hyperbolic-map-of-the-internet-will-save-it-from-COLLAPSE
